The usual combinatorial model for the 0-Hecke algebra of the symmetric group
is to consider the algebra (or monoid) generated by the bubble sort operators.
This construction generalizes to any finite Coxeter group W. The authors
previously introduced the Hecke group algebra, constructed as the algebra
generated simultaneously by the bubble sort and antisort operators, and
described its representation theory.

We provide the unique affine crystal structure for type E_6^{(1)}
Kirillov-Reshetikhin crystals corresponding to the multiples of fundamental
weights s Lambda_1, s Lambda_2, and s Lambda_6 for all s \geq 1 (in Bourbaki's
labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods
introduce a generalized tableaux model for classical highest weight crystals of
type E and use the order three automorphism of the affine E_6^{(1)} Dynkin
diagram.

We construct the Schubert basis of the torus-equivariant K-homology of the
affine Grassmannian of a simple algebraic group G, using the K-theoretic
NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a
construction of Peterson in equivariant homology.